Linear-Response TDDFTin Frequency-Reciprocal space

on a Plane-Waves basis:the DP (Dielectric Properties) code

Valerio Olevano,

Lucia Reining, Francesco Sottile and Silvana Botti

September 1, 2004

Why and Wherethe Frequency-Reciprocal space

could be convenient

• Reciprocal Space⇒ Infinite Periodic Systems (Bulk, but also Surfaces,

Wires, Tubes with the use of Supercells)

• Frequency Space ⇒ Spectra

Outlook

• Motivation

• TDDFT

• Linear Response TDDFT

• Frequency-Reciprocal space TDDFT

• TDDFT on a PW basis: the DP code

• Approximations and Results

Optical Absorption

2.5 3 3.5 4 4.5 5 5.5

ω [eV]

0

10

20

30

40

50

Im ε

(ω)

EXP

SiliconOptical Absorption

Lautenschlager etal., PRB 36, 4821(1987).

Aspnes and Studna,PRB 27, 985 (1983).

Refraction Indices

1.0 2.0 3.0 4.0 5.0 6.0 7.0

ω [eV]

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

n, k

n EXPk EXP

SiliconRefraction Indices

Reflectance

1 2 3 4 5 6 7

ω [eV]

0.3

0.4

0.5

0.6

0.7

0.8

R

EXP

SiliconReflectance

Aspnes and Studna,PRB 27, 985 (1983).

Stiebling, Z. Phys. B31, 355 (1978).

Energy-Loss (EELS)

10 15 20 25

energy loss, ω [eV]

0.0

1.0

2.0

3.0

4.0

5.0

- Im

ε-1

(ω)

EXP

SiliconEELS

Inelastic X-ray Scattering Spectroscopy (IXXS)

0.0 20.0 40.0 60.0

ω [eV]

0.0

0.2

0.4

-Im ε

-1(q

,ω)

EXP

SiliconIXSS Dynamic Structure Factor, q = 1.25 a.u. || [111]

Sturm et al., PRL 48,1425 (1982).

Schulke andKaprolat, PRL 67,879 (1991).

Coherent Inelastic Scattering Spectroscopy (CIXS)

0.0 10.0 20.0 30.0 40.0

ω [eV]

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

- Im

ε-1

(q,q

+G,ω

)

EXP

SiliconCIXS, q = (0.83,0.33,0.33) 2π/a, G = (-1,-1,-1) 2π/a

Problems for the Theory

• Reproduce Experimental Spectra

• Offer Reference Spectra to the Experiment

• Predict Optical and Dielectric Properties ⇒ Theoretical Spectroscopy

Facility

Accessed Observable:the Macroscopic Dielectric Function εM(q, ω)

ε∞ = εM(q = 0, ω = 0) dielectric constant

ABS = Im εM(q = 0, ω) optical absorption

EEL = −Im1

εM(q, ω)energy loss

What is the TDDFT?

• TDDFT is an extension of DFT; it is a DFT with time-dependent

external potential:

v(r) → v(r, t)

• Milestones of TDDFT:

– Runge, Gross (1984): rigorous basis of TDDFT.

– Gross, Kohn (1985): TDDFT in Linear Response.

DFT vs TDDFT

Hohenberg-Kohn:

v(r) ⇔ ρ(r)Runge-Gross:

v(r, t) ⇔ ρ(r, t)

The Total Energy:

〈Φ|H|Φ〉 = E[ρ]The Action:∫ t1

t0dt 〈Φ(t)|i ∂

∂t − H(t)|Φ(t)〉 = A[ρ]

are unique functionals of the density.

The stationary points of:

the Total EnergyδE[ρ]δρ(r) = 0

the Action:δA[ρ]δρ(r,t) = 0

give the exact density of the system:

ρ(r) ρ(r, t)

DFT vs TDDFT

Kohn-Sham:ρ(r)=

PNi=1|φKS

i (r)|2

vKS(r)=v(r)+R

dr′ ρ(r′)|r−r′|+

δExc[ρ]δρ(r)

HKS(r)φKSi (r)=εKS

i φKSi (r)

Runge-Gross:ρ(r,t)=

PNi=1|φKS

i (r,t)|2

vKS(r,t)=v(r,t)+R

dr′ ρ(r′,t)|r−r′|+

δAxc[ρ]δρ(r,t)

i ∂∂tφ

KSi (r,t)=HKS(r,t)φKS

i (r,t)

TDDFT in Linear ResponseGross and Kohn (1985)

If:

vext(r, t) = vext(r) + δvext(r, t)

with:

δvext(r, t) � vext(r)

then:

TDDFT = DFT + Linear Response(to the time-dependent perturbation δvext)

Hohenberg-Kohn Theoremfor Linear Response TDDFT

DFT: vext(r) ⇔ ρ(r)

TDDFT: vext(r, t) ⇔ ρ(r, t) Runge-Gross theorem

LR-TDDFT: vext(r) + δvext(r, t) ⇔ ρ(r) + δρ(r, t)⇓

δvext(r, t) ⇔ δρ(r, t)

Linear Response TDDFTcalculation scheme

1. Ordinary DFT calculation:

vext(r) ⇒ ρ(r), εKS, φKS(r)

2. Linear Response calculation:

δvext(r, t) ⇒ δρ(r, t)

Polarizability χ

δvext external perturbation

δρ induced density

Definition of the polarizability χ:

δρ = χδvext

Variation in the Total Potential

The variation in the density induces a variation in the Hartree and in the exchange-

correlation potentials which screen the external perturbation:

δvH =δvH

δρδρ = vcδρ

δvxc =δvxc

δρδρ = fxcδρ

so that the variation in the total potential (external + screening) is

δvtot = δvext + δvH + δvxc

Exchange-Correlation Kernel fxc

The exchange-correlation kernel is defined:

fxcdef=

δvxc

δρ

LR-TDDFT Kohn-Sham scheme:the Independent Particle Polarizability χ(0)

Let’s introduce a ficticious s, Kohn-Sham non-interacting system such that:

δρs = δρ

Then, instead of calculating χ, we can more easily calculate the polarizability χ(0) (also

χs or χKS) of this non-interacting system, called independent particle polarizability

and defined:

δρ = χ(0)δvtot

Independent Particle Polarizability χ(0)

By variation δvtot (= δvs = δvKS) of the Kohn-Sham equations, one obtains theLinear Response variation of the density δρ and then an expression for χ(0) in termsof the Kohn-Sham energies and wavefunctions:

χ(0)(r, r′, ω) = 2∑i 6=j

(fi − fj)φi(r)φ∗j(r)φ

∗i (r

′)φj(r′)εi − εj − ω − iη

(Adler and Wiser)

In Frequency Reciprocal space:

χ(0)GG′(q, ω) = 2

∑i 6=j

(fi − fj)〈φj|e−i(q+G)r|φi〉〈φi|ei(q+G′)r|φj〉

εi − εj − ω − iη

χ as a function of χ(0)

From:{δρ = χδvext

δρ = χ(0)δvtot

the polarizability in term of the independent particle polarizability is:

χ = (1− χ(0)vc − χ(0)fxc)−1χ(0)

Dielectric Function ε

Definition of the dielectric function:

δvtot = ε−1δvext

ε−1 = 1 + vcχ

In a periodic system it has this form:

εGG′(q, ω)

Macroscopic Dielectric Function

Definition:

εM(q, ω) def=1

ε−100 (q, ω)

Approximation: Neglecting Local Fields:

εNLFM (q, ω) = ε00(q, ω)

Macroscopic dielectric constant:

ε∞ = limq→0

εM(q, ω = 0)

Local Fields

Effect of non-diagonal elements (inhomogeneities):

δvtotG =

∑G′

ε−1GG′ δv

extG′

LR-TDDFT Calculation SchemeResume

ρDFT

"""b"b

"b"b

"b

φDFTKS

##GGGGGGGG

|| |<|<

|<|<

|<

εDFTKS

||xxxxxxxx

fxc

++

χ(0)

��

χ

��

ε

��

εM

the code

• Definition: Linear-Response TDDFT code in Frequency-Reciprocal space on a PW basis

• Purposes: DP calculates Dielectric and Optical Properties (Absorption, Reflectivity, Refraction

indices, EELS, IXSS, CIXS, etc.) for Bulk systems, Surfaces, Cluster, Moleculs, Atoms (through

Supercells) made of Insulator, Semiconductor and Metallic elements.

• Approximations: RPA, ALDA, GW-RPA, LRC, non-local kernels, Mapping Theory, etc., with

and without Local Fields (LF)

• Languages: Fortran90 with C insertions (shell, parser, some libraries); Vectorialized and Partially

Parallelized (MPI)

• Machines: PC-Linux IFC, Compaq/HP True64, IBM AIX, SG IRIX, NEC SX5, Fujitsu

• Libraries: BLAS, Lapack, FFTW, CXML, ESSL, ASL, Nag, Goedecker, MFFT

• Interfaces: LSI-CP, ABINIT, PWSCF, FHIMD

DP package, Copyright 1998-2004 Valerio Olevano, Lucia Reining, Francesco Sottile, CNRS.

DP Flow Diagram

εKSi

��

φKSi (G)

FFT−1��

interface

φKSi (r)

��

ρij(r) = φ∗i (r)φj(r)

FFT��

ρij(G)

��

χ(0)GG′(q, ω) =

∑ij(fi − fj)

ρij(G)ρ∗ij(G′)

εi−εj−ω

INV��

fxc// χGG′(q, ω)

��

ε−1GG′(q, ω)

sshhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

++VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV

εLFM (q, ω) εNLF

M (q, ω)

DP Tricks

If we only need ε−100 = 1− v0χ00, that is only χ00

then, instead of solving (inverting a full matrix):

χGG′ =(1− χ(0)vc − χ(0)fxc

)−1

GG′′χ

(0)G′′G′

we solve the linear system for only the first column of χG′0:(1− χ(0)vc − χ(0)fxc

)GG′

χG′0 = χ(0)G0

DP Performances:CPU scaling and Memory usage

• CPU scaling for χ(0): N2G ·Nk ·Nb ·Nr log Nr

• CPU scaling for ε−1: N2G ·Nω

• Memory occupation: N2G ·Nω + Nr ·Nk ·Nb [sizeofcomplex]

Exchange-Correlation Kernel fxc

and its Approximations

Definition of the exchange-correlation kernel:

fxcdef=

δvxc

δρ

RPA Approximation

Random Phase Approximation (RPA) = Neglect of the exchange-correlation

effects:

RPA: fxc = 0

εRPA = 1− vcχ(0)

ALDA Approximation (TDLDA)

Adiabatic Local Density Approximation:

ALDA: fALDAxc =

δvLDAxc

δρ

∣∣∣∣ω=0

fALDAxc (r, r′) = A(r)δ(r, r′) local in r-space

fALDAxc GG′(q) = B(G−G′)

Results on EELS

10 20 30 40energy [eV]

0

2

loss

fun

ctio

n

RPA LFRPA NLFEXPTDLDA

GraphiteEELS

Θ = 90ο

Θ = 45ο

Θ = 30ο

Θ = 0ο

Θ = 77ο

in plane

out of plane

A. Marinopoulos et al., Phys. Rev. B 69, 245419 (2004).

Results on IXSS and CIXS

0 10 20 30 40 50 60

ω [eV]

0.0

0.2

0.4

0.6

-Im ε

-1(q

,ω)

EXPRPATDLDA

SiliconIXSS Dynamic Structure Factor, q = 1.25 a.u. || [111]

0.0 10.0 20.0 30.0 40.0

ω [eV]

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

- Im

ε-1

(q,q

+G,ω

)RPATDLDAEXP

SiliconCIXS, q = (0.83,0.33,0.33) 2π/a, G = (-1,-1,-1) 2π/a

V. Olevano, PhD thesis (1999).

Conclusions on EELS

• RPA is good but with LF (Local Fields)

• ALDA does not improve unambigously

Results on ABS

3 4 5 6

ω [eV]

0

10

20

30

40

50

60

ε 2(ω)

EXPRPATDLDARPA NLF

SiliconOptical Absorption

LRC Approximation

Long-Range Contribution only:

fLRCxc = − α

(q + G)2

α = 4.6ε−1∞ − 0.2

Results on ABS

3 4 5 6

ω [eV]

0

10

20

30

40

50

60

ε 2(ω)

EXPRPATDLDATDDFT LRC

SiliconOptical Absorption

L. Reining et al., Phys. Rev. Lett. 88, 066404 (2002).

Mapping Theory (MT)

Mapping BSE on TDDFT:

fMPxc [{φi}, {εi}]

χ = χ(0)(χ(0) − χ(0)vcχ

(0) − T [{φi}, {εi}])−1

χ(0)

T = χ(0)fxcχ(0)

Results on Solid Argon ABS

11 12 13 14 150

5

10

15Exp.BSETDLDAT2

F. Sottile et al., to be published

Conclusions on Optical Properties

• RPA and ALDA fail

• LRC already reproduces small Excitonic Effects

• Mapping Theory should be used for Strong Excitons

DP licence

• DP costed 7 years human full-time to the authors: is GNU/GPL going to reducethe charge on the central?

• developping software is a job in itself!

• who is going to pay for this? are private companies interested in this software? notyet!

• if the scientific community is interested and recognize the importance, then it musthire people to develop these codes!

• International, European or even National institutions must be setup to managescientific codes interesting for the whole international scientific community.

DP licence and ETSF

IF ( )

THEN

= OpenSource GNU/GPL